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Difference between revisions of "Poly-nilpotent group"

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A group possessing a finite [[Normal series|normal series]] with nilpotent factors; such a series is called poly-nilpotent. The length of the shortest poly-nilpotent series of a poly-nilpotent group is called its poly-nilpotent length. The class of all poly-nilpotent groups coincides with the class of all solvable groups (cf. [[Solvable group|Solvable group]]); however, in general the poly-nilpotent length is less than the solvable length. A poly-nilpotent group of length 2 is called meta-nilpotent.
 
A group possessing a finite [[Normal series|normal series]] with nilpotent factors; such a series is called poly-nilpotent. The length of the shortest poly-nilpotent series of a poly-nilpotent group is called its poly-nilpotent length. The class of all poly-nilpotent groups coincides with the class of all solvable groups (cf. [[Solvable group|Solvable group]]); however, in general the poly-nilpotent length is less than the solvable length. A poly-nilpotent group of length 2 is called meta-nilpotent.
  
All groups having (an increasing) poly-nilpotent series of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073520/p0735201.png" /> whose factors in increasing order have nilpotent classes not exceeding the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073520/p0735202.png" />, respectively, form a variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073520/p0735203.png" />, which is the product of nilpotent varieties:
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All groups having (an increasing) poly-nilpotent series of length $l$ whose factors in increasing order have nilpotent classes not exceeding the numbers $c_1,\dots,c_l$, respectively, form a variety $\mathfrak M$, which is the product of nilpotent varieties:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073520/p0735204.png" /></td> </tr></table>
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$$\mathfrak M=\mathfrak N_{c_1}\dots\mathfrak N_{c_l}$$
  
(see [[Variety of groups|Variety of groups]]). The free groups of such a variety are called free poly-nilpotent groups. Of particular interest are the varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073520/p0735205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073520/p0735206.png" />. The first of them contains all connected solvable Lie groups; in the second, all finitely-generated groups are finitely approximable and satisfy the maximum condition for normal subgroups.
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(see [[Variety of groups|Variety of groups]]). The free groups of such a variety are called free poly-nilpotent groups. Of particular interest are the varieties $\mathfrak N_c\mathfrak A$ and $\mathfrak A\mathfrak N_c$. The first of them contains all connected solvable Lie groups; in the second, all finitely-generated groups are finitely approximable and satisfy the maximum condition for normal subgroups.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''1–2''' , Chelsea  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Neumann,  "Varieties of groups" , Springer  (1967)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''1–2''' , Chelsea  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Neumann,  "Varieties of groups" , Springer  (1967)</TD></TR></table>

Latest revision as of 16:12, 4 October 2014

A group possessing a finite normal series with nilpotent factors; such a series is called poly-nilpotent. The length of the shortest poly-nilpotent series of a poly-nilpotent group is called its poly-nilpotent length. The class of all poly-nilpotent groups coincides with the class of all solvable groups (cf. Solvable group); however, in general the poly-nilpotent length is less than the solvable length. A poly-nilpotent group of length 2 is called meta-nilpotent.

All groups having (an increasing) poly-nilpotent series of length $l$ whose factors in increasing order have nilpotent classes not exceeding the numbers $c_1,\dots,c_l$, respectively, form a variety $\mathfrak M$, which is the product of nilpotent varieties:

$$\mathfrak M=\mathfrak N_{c_1}\dots\mathfrak N_{c_l}$$

(see Variety of groups). The free groups of such a variety are called free poly-nilpotent groups. Of particular interest are the varieties $\mathfrak N_c\mathfrak A$ and $\mathfrak A\mathfrak N_c$. The first of them contains all connected solvable Lie groups; in the second, all finitely-generated groups are finitely approximable and satisfy the maximum condition for normal subgroups.

References

[1] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1960) (Translated from Russian)
[2] H. Neumann, "Varieties of groups" , Springer (1967)
How to Cite This Entry:
Poly-nilpotent group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poly-nilpotent_group&oldid=19220
This article was adapted from an original article by A.L. Shmel'kin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article